Simplifying Rational Expressions - Metropolitan Community College
Rational Expressions
A quotient of two integers, , where 0, is called a rational expression.
Some
examples
of
rational
expressions
are
,
,
,
and
.
When
4, the denominator of the
expression
becomes
0
and
the
expression is
meaningless.
Mathematicians state this fact by saying that
the
expression
is
undefined
when
4.
One can see that the value
,
makes
the
expression
undefined. On the other hand, when any real number is substituted into the expression , the answer is
always a real number. There are no values for which this expression is undefined.
EXAMPLE Solution
Determine the value or values of the variable for which the rational expression is defined.
a)
b)
a) Determine the value or values of x that make 2x ? 5 equal to 0 and exclude these. This can be done by setting 2x ? 5 equal to 0 and solving the equation for x.
2 5 0
2 5
Do
not consider
when
considering the rational expression
.
This
expression
is
defined for all real numbers except . Sometimes to shorten the answer it is written as
.
b) To determine the value or values that are excluded, set the denominator equal to zero and
solve the equation for the variable.
6 7 0
7 1 0
7 0 or 1 0
7
1
Therefore, do not consider the values 7 or 1 when considering the rational
expression . Both 7 and 1make the denominator zero. This is defined for all real numbers except 7 and 1. Thus, 7 and 1.
SIGNS OF A FRACTION
Notice:
Generally, a fraction is not written with a negative denominator.
For
example,
the
expression
would
be
written
as
either
or
.
The
expression
can
be
written
since
4
4 or 4.
Other examples of equivalent fractions:
SIMPLIFYING RATIONAL EXPRESSIONS
A rational expression is simplified or reduced to its lowest terms when the numerator and denominator have
no common factors other than 1.
The
fraction
is
not
simplified
because
9
and
12
both
contain
the
common
factor 3. When the 3 is factored out, the simplified fraction is .
? ?
The
rational
expression
is
not
simplified
because
both
the
numerator
and
denominator
have
a
common
factor, b. to simplify this expression, factor b from each term in the numerator, then divide it out.
Thus,
becomes
when
simplified.
To Simplify Rational Expressions 1. Factor both the numerator and denominator as completely as possible. 2. Divide out any factors common to both the numerator and denominator.
Example 1
Simplify
Solution Factor the greatest common factor, 5, from each term in the numerator. Since 5 is a factor common to both the numerator and denominator, divide it out.
Example 2
Simplify
?
Solution Factor the numerator; then divide out the common factor.
1
Example 3
Simplify
Solution Factor the numerator; then divide out common factors.
=4
Example 4
Simplify
Solution Factor both the numerator and denominator, then divide out common factors.
Example 5
Simplify
Solution Factor both the numerator and denominator, then divide out common factors.
Example 6
Simplify
Solution Factor both the numerator and denominator, then divide out common factors.
Example 7
Simplify
Solution Factor both numerator and denominator, then divide out common factors.
Consider the expression , a common student error is to attempt to cancel the x or the 3 or both x and 3 appearing in this expression.
This
is
WRONG!
does
not
equal
2
It is WRONG because factors are not being reduced. Evaluating this expression for an easy value, such as
1, would show that the illustrated cancellations are WRONG.
If
1,
becomes
.
Remember: Only common factors can be divided out from expressions.
5
In the denominator of the example on the left, 4, the 4 and x are factors since they are multiplied together. The 4 and the x are also both factors of the numerator 20, since 20 can be written 4 ? ? 5 ? .
Some students incorrectly divide out terms. In the expression , the x and ?4 are terms of the denominator, not factors, and therefore cannot be divided out.
Recall that when -1 is factored from a polynomial, the sign of each term in the polynomial changes.
EXAMPLES:
3 5 13 5 3 5
6 2 16 2 2 6
Example 8
Simplify
Solution Since each term in the numerator differs only in sign from its like term in the denominator, factor-1 from each term in the denominator.
1
Example 9
Simplify
Solution
4 1
ADDITIONAL EXERCISES
Determine the value or values of the variables for which the expression is defined.
1.
2.
3.
Simplify
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Answers
1. 3, 4
2.
3. 6, 1
4.
0,
5. 6, 2
6.
,
0
7.
8.
9.
10. 3
11. 2
12. ? 7
13. ? 2
14.
15.
16.
17.
18. ? 7
19.
20.
?
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